CN105182744B - Anti-interference control method for nanometer positioning system - Google Patents

Abstract

The invention discloses an anti-interference control method for a nanometer positioning system. The anti-interference control method comprises the steps that: an extended state observer based on error transformation receives system output y and controlled quantity u, and calculates an interference estimated value and a state estimated value according to the controlled quantity u and the system output y; the observer transmits the state estimated value to a controller, and the controller calculates a control law based on expected system output y and the state estimated value; the controlled quantity of the nanometer positioning system is obtained through calculation according to the control law and the interference estimated value; and the system controlled quantity is input to the nanometer positioning system. The anti-interference control method utilizes excellent performance of nonlinear functions, clarifies physical significance of parameters in the nonlinear functions, has high operability in parameter setting, and reduces the setting difficulty.

Description

The anti-interference control method of axis Nano-positioners

Technical field

The present invention relates to the anti-interference control in the Advanced Control Techniques field of axis Nano-positioners, particularly axis Nano-positioners Technology processed, is specifically the Auto Disturbances Rejection Control Technique based on error transform in axis Nano-positioners.

Background technology

Precision positioning technology is atomic force microscope, PSTM, circuit optical fiber be aligned, biological microscope surgery One of key technology in the modern sciences and engineering such as operation, biological microtechnic and accurate assembling.Driver is to produce to drive The device of effect or device, are the important component parts in Precision Position Location System, its performance directly affects the positioning of alignment system Precision.Piezoelectric actuator becomes because having the advantages such as high resolution, small volume, High power output, frequency be high, heating is little and response is fast The first-selection of Precision Position Location System.Although the good characteristic of piezoelectric actuator drastically increases the performance of precisely locating platform, It is the positioning precision that the intrinsic Hysteresis Nonlinear characteristic of piezoelectric actuator significantly reduces system, or even make system unstability.

For overcoming sluggish precision system positioning performance being affected, improving location control, control technology becomes precision positioning One of core technology of technology.Chinese scholars are paid and are a lot made great efforts it is proposed that different control methods.Sum up and substantially can divide For three classes, it is respectively：Electric charge drives control, the control based on the compensation of sluggish inversion model and the direct controlling party of piezoelectric actuator Method.

Effectively reduce the Hysteresis Nonlinear of piezoelectric actuator using electric charge driving energy, but charge amplifier design complexity, High cost, unfavorable its is extensively applied.

The control method being compensated based on sluggish inversion model, its core is the sluggish inversion model of construction to offset sluggishness to systematicness The impact of energy.At present, main sluggish inversion model has：Preisach inversion model, KP inversion model, PI inversion model and nerve network reverse Model etc..Using sluggish inversion model, mainly there are two class control structures：One class is that sluggish inversion model is directly connected with controlled device To compensate sluggish control structure；Another kind of it is the feedforward compensation of sluggish inversion model and complex controll knot that feedback control combines Structure.For example：Guo-Ying Gu etc. describes the hesitation of piezoelectric actuator with a kind of improved PI Hysteresis Model, then will Sluggish inversion model is connected with controlled device with the sluggish impact of real-Time Compensation；Hui Tang etc. is directed to and is described with Preisach model Sluggishness, first carry out feedforward compensation, then carry out closed loop control using non-linearity PID, thus improving the performance of whole system.Can See, the parsing all relying on Hysteresis Model based on the control method that sluggish inversion model compensates is inverse.However, Hysteresis Model is relatively multiple Miscellaneous, the degree of accuracy of Hysteresis Model directly influences the precision of sluggish inversion model and the sluggish inverse effect compensating；Additionally, sluggish inverse mould Type whether there is and the complexity etc. of inversion model all can affect the control performance of system.

For avoiding above-mentioned deficiency, domestic and international expert proposes the direct control method without inversion model.The method is not set up Sluggish inversion model, mainly obtains control action according to the deviation of the desired output of system and reality output, reaches control targe, with PID control is representative.However, PID control passively eliminates deviation it is impossible to the control meeting hi-Fix requires.

In recent years, factor system being deviateed desired output is considered as disturbing, and estimates on one's own initiative and compensating disturbance is to obtain the phase The disturbance rejection control method (referred to as active disturbance rejection control) hoping output is gradually by controlling boundary to be paid attention to, and obtains in precise Positioning Control To apply.In active disturbance rejection control technology, a typical class is Active Disturbance Rejection Control.It is proposed by system institute Mr. Han Jingqing, its Core concept is with integration series relationship simple between system output and control input as standard type, differing from system dynamics The part of standard type is considered as " total disturbance " (disturb in inclusion and disturb outward), with extended state observer as means, estimates in real time simultaneously Eliminate " total disturbance ", thus the integration tandem type being reduced to standard full of disturbance, uncertain and nonlinear controlled device. But, it emphasizes to utilize nonlinear characteristic, and parameter is more, and parameter does not have clear and definite physical significance, is largely dependent upon Experience, has larger difficulty of adjusting.For solving this problem, professor Gao Zhiqiang proposes that a kind of parameter is less, easily adjust Linear active disturbance rejection controller.Though linear active disturbance rejection controller solves the problems, such as parameter tuning, reduce to a certain extent Control performance.

Content of the invention

The invention aims to overcoming the shortcomings of that linear active disturbance rejection controls, invent a kind of antidisturbance control technology, it Both utilize nonlinear function excellent properties, and in clear and definite nonlinear function each parameter physical significance, so that parameter tuning is had Stronger operability, reduces its difficulty of adjusting；Especially, it can be applied to axis Nano-positioners.

The present invention is achieved through the following technical solutions：

For making full use of nonlinear function characteristic, improve control performance, by default capabilities function, make observer error warp After crossing non-linear function transformation, it is limited to all the time in certain scope, so that observer is more accurately estimated and eliminate total disturbance.

For realizing the purpose of the present invention, employ the following technical solutions and be achieved：

A kind of anti-interference control method of axis Nano-positioners, comprises the following steps：

Extended state observer based on error transform receives output y and controlled quentity controlled variable u of described system, is calculated according to y and u Interference estimate and state estimation；

State estimation is sent to controller by described observer, and controller exports y according to desired system_dAnd state Estimated value calculates control law u₀；

Calculate controlled quentity controlled variable u of axis Nano-positioners according to described control law and interference estimate；

Described system control amount u is inputted described axis Nano-positioners.

The anti-interference control method of described axis Nano-positioners, wherein

Observer calculates interference estimate z as follows respectively₃With state estimation z₁,z₂

$\left\{\begin{array}{c}\frac{{\mathrm{dz}}_1\left(t\right)}{dt}=z_2\left(t\right)+{\mathrm{\β}}_1{e}_o\left(t\right)\\ \frac{{\mathrm{dz}}_2\left(t\right)}{dt}=z_3\left(t\right)+{\mathrm{\β}}_2\mathrm{\ρ}\left(t\right)S\left(e_o\right(t\left)\right)+b_{0}u\left(t\right)\\ \frac{{\mathrm{dz}}_3\left(t\right)}{dt}={\mathrm{\β}}_3\mathrm{\ρ}\left(t\right)S\left(e_o\right(t\left)\right)\end{array}\right.$

Control law u₀Computing formula as follows：u₀=k₁(y_d-z₁)-k₂z₂

Wherein：β₁、β₂、β₃Be it needs to be determined that parameter；The default capabilities function that ρ (t) is positive definite, successively decrease, anaplasia when t is Amount；S (x) be smooth, strictly increasing function；Observer estimated bias are e_o(t)=y-z₁；b₀Coefficient for controlled quentity controlled variable u；ω_cIt is control bandwidth.

The anti-interference control method of described axis Nano-positioners, wherein

Controller is calculated as follows controlled quentity controlled variable u： $u=\frac{u_0-z_3}{b_0}=\frac{k_1(y_d-z_1)-k_2{z}_2-z_3}{b_0}.$

The anti-interference control method of described axis Nano-positioners, wherein：And ρ (t) meets：

$-{\mathrm{\&delta;}}_1\mathrm{\&rho;}\left(t\right)<e\left(t\right)<{\mathrm{\&delta;}}_2\mathrm{\&rho;}\left(t\right),\&ForAll;t\&GreaterEqual;0$

Wherein：δ₁,δ₂For adjustable parameter, 0 ＜ δ₁,δ₂≤ 1 ,-δ₁ρ (0) and δ₂ρ (0) is respectively deviation e (t) after converting Lower bound and the upper bound.

The anti-interference control method of described axis Nano-positioners, wherein：S (x) meets following condition：

-δ₁＜ S (x) ＜ δ₂

$\underset{x\&RightArrow;\mathrm{\&infin;}}{\lim }S\left(x\right)={\mathrm{\&delta;}}_2,\underset{x\&RightArrow;-\mathrm{\&infin;}}{\lim }S\left(x\right)=-{\mathrm{\&delta;}}_1$

The anti-interference control method of described axis Nano-positioners, described axis Nano-positioners are the nanometer positioning of Piezoelectric Driving System.

Brief description

Fig. 1 is the system block diagram of the present invention；

Fig. 2 is the flow chart of the present invention；

Fig. 3 is prior-art illustration.

Specific embodiment

The structured flowchart of generally Active Disturbance Rejection Control is as shown in Figure 3.As shown in figure 3, y_dFor the desired output of system, u (t) table Show control law, y represents that system exports.Real-time estimation is simultaneously for extended state observer (Extended State Observer, ESO) Compensate the interference signal of impact system output, it is Active Disturbance Rejection Control (Active Disturbance Rejection Control, ADRC) core.Taking second order controlled system as a example, linear extended state observer (Linear Extended State Observer, LESO) state-space expression be:

$\left\{\begin{array}{c}\frac{{\mathrm{dz}}_1\left(t\right)}{dt}=z_2\left(t\right)+{\mathrm{\&beta;}}_1(y-z_1)\\ \frac{{\mathrm{dz}}_2\left(t\right)}{dt}=z_3\left(t\right)+{\mathrm{\&beta;}}_2(y-z_1)+b_{0}u\left(t\right)\\ \frac{{\mathrm{dz}}_3\left(t\right)}{dt}={\mathrm{\&beta;}}_3(y-z_1)\end{array}\right.$

Wherein：β₁、β₂、β₃Be it needs to be determined that parameter, z₁、z₂、z₃It is the state variable of extended state observer ESO, point Other tracking system exports y,And the total disturbance of system (being designated as f), b₀For the coefficient of controlled quentity controlled variable u, observer estimated bias are e_o(t) =y-z₁.

In fact, the convergence rate of observer estimated bias is heavily dependent on expansion state imitating to the estimation of disturbance Really, that is, z₃Estimation effect to f.If the estimated bias of observer are limited within the specific limits, will greatly be lifted The efficiency of interference estimated by observer, and then the control performance of lift system.That is, limiting the estimated bias of observer, carrying The convergence rate of high observer, actually will improve the effect that expansion state estimates disturbance, lift extended state observer Estimation performance, and then improve closed loop control performance.

The present invention passes through to introduce error transform, reduces observer estimated bias, thus lifting the estimation performance of observer, with Shi Tigao parameter beta₁、β₂、β₃Adjustable extent, reduce parameter tuning difficulty.

For limit error, default capabilities function ρ (t) introduce positive definite, successively decreasing (t is time variable), haveAnd ρ (t) meets：

$-{\mathrm{\&delta;}}_1\mathrm{\&rho;}\left(t\right)<e\left(t\right)<{\mathrm{\&delta;}}_2\mathrm{\&rho;}\left(t\right),\&ForAll;t\&GreaterEqual;0$

Wherein：δ₁,δ₂For adjustable parameter, 0 ＜ δ₁,δ₂≤ 1 ,-δ₁ρ (0) and δ₂ρ (0) is respectively deviation e (t) after converting Lower bound and the upper bound.The present invention selects

ρ (t)=(ρ₀-ρ_∞)exp(-lt)+ρ_∞

Wherein：ρ₀＞ ρ_∞, l ＞ 0, ρ₀=ρ (0) is the maximum of allowable error, ρ_∞It is that system reaches stable time error Big value, the rate of decay of l determining function ρ (t).

In order to meet conditions above, present invention introduces smooth, strictly increasing function S (x), S (x) meets following bar Part：

-δ₁＜ S (x) ＜ δ₂

$\underset{x\&RightArrow;\mathrm{\&infin;}}{\lim }S\left(x\right)={\mathrm{\&delta;}}_2,\underset{x\&RightArrow;-\mathrm{\&infin;}}{\lim }S\left(x\right)=-{\mathrm{\&delta;}}_1$

By the expression formula that function above can obtain error transform it is：

E (t)=ρ (t) S (x)

E (t) after error transform is substituted in LESO, the extended state observer based on error transform can be obtained The state-space expression of (Error Transformation based Extended State Observer, ETESO) is：

$\left\{\begin{array}{c}\frac{{\mathrm{dz}}_1\left(t\right)}{dt}=z_2\left(t\right)+{\mathrm{\&beta;}}_1{e}_o\left(t\right)\\ \frac{{\mathrm{dz}}_2\left(t\right)}{dt}=z_3\left(t\right)+{\mathrm{\&beta;}}_2\mathrm{\&rho;}\left(t\right)S\left(e_o\right(t\left)\right)+b_{0}u\left(t\right)\\ \frac{{\mathrm{dz}}_3\left(t\right)}{dt}={\mathrm{\&beta;}}_3\mathrm{\&rho;}\left(t\right)S\left(e_o\right(t\left)\right)\end{array}\right.$

The present invention selects $S\left(x\right)=\frac{{\mathrm{\&delta;}}_2\exp \left(x\right)-{\mathrm{\&delta;}}_1\exp (-x)}{\exp \left(x\right)+\exp (-x)}.$

For ease of parameter tuning, by the adjustable parameter β in ETESO_iChoosing method be defined as：

$s^n+\underset{i=1}{\overset{n}{\&Sigma;}}{\mathrm{\&beta;}}_i{s}^{n-i}={(s+{\mathrm{\&omega;}}_o)}^n$

Wherein ω_oFor observation bandwidth, n is integer, and s is Laplace operator.

Three rank ETESO are had

The controller of ADRC partly each parameter k_iChoosing method be defined as：

$s^n+\underset{i=1}{\overset{n}{\&Sigma;}}{k}_i{s}^{n-i}={(s+{\mathrm{\&omega;}}_c)}^n$

Wherein ω_cFor control bandwidth, s is Laplace operator.

Second order ADRC is had

So adjustable parameter of ADRC is just changed into：Control bandwidth ω_c, observation bandwidth ω_o；Error transform constant：Allowable error Maximum ρ₀, the maximum ρ of system stability time error_∞, rate of decay l of ρ (t)；The bound δ of deviation after conversion₁,δ₂.

These parameters all have specific physical significance, have so both remained the performance of nonlinear function, make parameter whole again Surely there is stronger operability.

The present invention proposes a kind of improved design project of Active Disturbance Rejection Control, compared with existing Auto Disturbances Rejection Control Technique, Following significant advantage is had based on the Auto Disturbances Rejection Control Technique of error transform：

1. remain the premium properties of nonlinear function, contribute to the lifting of control performance；

By introducing nonlinearity erron transforming function transformation function, using the good characteristic of nonlinear function, it is to avoid linear active disturbance rejection Control the deficiency simply utilizing observed deviation loss observation performance.

2. each parameter of nonlinear function has clear and definite physical significance, conveniently adjusts；

The nonlinearity erron transforming function transformation function introducing, its parameter all has clear and definite physical significance, can be according to actual control The demand of system is adjusted, workable.

3., for the axis Nano-positioners of Piezoelectric Ceramic, cause because the sluggishness of piezoelectric actuator, creep etc. are non-linear Systematic function reduce and can be eliminated by the Active Disturbance Rejection Control based on error transform, and control performance can be according to production technology Demand is adjusted.

Below in conjunction with accompanying drawing, the present invention is described in further detail：

As shown in figure 1, the control system of the present invention is the Active Disturbance Rejection Control system based on error transform, its core is Based on the extended state observer of error transform, i.e. ETESO.In implementation process, first, calculating observation device exports to system Estimate the deviation and system real output value between, that is, observed deviation e_o(t)=y-z₁；Afterwards, this observed deviation is carried out Error transform, the expression formula of error transform is：

E (t)=ρ (t) S (x)

Using observed deviation as the input variable of error transform function, after output as error transform, it is used for expansion state The observed deviation of observer, i.e. e (t)=ρ (t) S (e_o(t)).Non-linear expansion state can be realized using this mapping fault to see Survey device, the performance of lifting Interference Estimation, and then lift control performance.

In parameter tuning, according to system demand for control, set following parameter：

■ control bandwidth ω_c, observation bandwidth ω_o；

The maximum ρ of ■ allowable error₀, the maximum ρ of system stability time error_∞, rate of decay l of ρ (t)；

The bound δ of deviation after ■ conversion₁,δ₂.

Decide whether to change relevant parameter according to running effect and control requirement.

Fig. 2 is the Active Disturbance Rejection Control design flow diagram based on error transform, specific as follows：

According to the extended state observer structure based on error transform, build the expansion state observation based on error transform Device；Meanwhile, build controller；Afterwards, require to set each control parameter according to control；Control action is put on Piezoelectric Driving Nanopositioning Mechanism；Then, observe control effect, require if meeting and controlling, complete design, be unsatisfactory for control and require then to return Return and readjust control parameter, require until meeting and controlling.

Claims (5)

1. a kind of anti-interference control method of axis Nano-positioners, comprises the following steps：

Extended state observer based on error transform receives output y and controlled quentity controlled variable u of described system, calculates interference according to y and u Estimated value and state estimation；

State estimation is sent to controller by described observer, and controller exports y according to desired system_dAnd state estimation Value calculates control law u₀；

Calculate controlled quentity controlled variable u of axis Nano-positioners according to described control law and interference estimate；

Described controlled quentity controlled variable u is inputted described axis Nano-positioners；

It is characterized in that：Observer calculates interference estimate z as follows respectively₃With state estimation z₁,z₂

$\left\{\begin{array}{c}\frac{{\mathrm{dz}}_1\left(t\right)}{dt}=z_2\left(t\right)+{\mathrm{\&beta;}}_1{e}_o\left(t\right)\\ \frac{{\mathrm{dz}}_2\left(t\right)}{dt}=z_3\left(t\right)+{\mathrm{\&beta;}}_2\mathrm{\&rho;}\left(t\right)S\left(e_o\right(t\left)\right)+b_{0}u\left(t\right)\\ \frac{{\mathrm{dz}}_3\left(t\right)}{dt}={\mathrm{\&beta;}}_3\mathrm{\&rho;}\left(t\right)S\left(e_o\right(t\left)\right)\end{array}\right.$

Control law u₀Computing formula as follows：u₀=k₁(y_d-z₁)-k₂z₂

Wherein：β₁、β₂、β₃Be it needs to be determined that parameter；The default capabilities function that ρ (t) is positive definite, successively decrease, t is time variable；S (x) be smooth, strictly increasing function；Observer estimated bias are e_o(t)=y-z₁；b₀Coefficient for controlled quentity controlled variable u；k₁=2 ω_c,ω_cIt is control bandwidth.

2. the anti-interference control method of axis Nano-positioners according to claim 1, wherein controller is calculated as follows Controlled quentity controlled variable u：

3. the anti-interference control method of axis Nano-positioners according to claim 1 and 2, wherein： And ρ (t) meets：

$-{\mathrm{\&delta;}}_1\mathrm{\&rho;}\left(t\right)<e\left(t\right)<{\mathrm{\&delta;}}_2\mathrm{\&rho;}\left(t\right),\&ForAll;t\&GreaterEqual;0$

Wherein：δ₁,δ₂For adjustable parameter, 0 ＜ δ₁,δ₂≤ 1 ,-δ₁ρ (0) and δ₂ρ (0) is respectively the lower bound of deviation e (t) after converting And the upper bound.

4. the anti-interference control method of axis Nano-positioners according to claim 3, wherein：S (x) meets following condition：

-δ₁＜ S (x) ＜ δ₂

$\underset{x\&RightArrow;\mathrm{\&infin;}}{\lim }S\left(x\right)={\mathrm{\&delta;}}_2,\underset{x\&RightArrow;-\mathrm{\&infin;}}{\lim }S\left(x\right)=-{\mathrm{\&delta;}}_1.$

5. the anti-interference control method according to the described axis Nano-positioners of one of claim 1-2, described axis Nano-positioners are The axis Nano-positioners of Piezoelectric Driving.